Physics in Motion

Axis Theorems

Parallel-Axis Theorem

If the moment of inertia about an axis passing through the center of mass is \( I_{CM} \), then the moment of intertia about a parallel axis displaced by a distance \( d \) from the cener of mass axis is:

$$ I = I_{CM} + M d^2 $$

Perpendicular-Axis Theorem

The moment of inertia of a plane object about an axis perpendicular to the plane is equal to the sum of the moments of inertia about any two perpendicular axes in the plane. Thus is the \( x \) and \( y \) axes are in the plane,

$$ I_z = I_x + I_y $$